Modus Tollens — Definition, Formula & Examples
Modus tollens is a rule of logical reasoning that says: if a conditional statement is true and its conclusion is false, then its hypothesis must also be false. In short, denying the conclusion forces you to deny the hypothesis.
Modus tollens (Latin for "method of denying") is the valid inference rule stating that from the premises and , one may conclude .
Key Formula
Where:
- = The hypothesis (antecedent) of the conditional
- = The conclusion (consequent) of the conditional
- = Logical negation ("not")
How It Works
Start with a conditional statement of the form "if , then ." Next, establish that is false (). Because the conditional guarantees whenever is true, the failure of means cannot be true either. You therefore conclude . Notice that modus tollens is logically equivalent to reasoning via the contrapositive: is the same as .
Example
Problem: Given: (1) If a shape is a square, then it has exactly four sides. (2) A particular shape does not have exactly four sides. What can you conclude?
Identify the conditional: Let P = "the shape is a square" and Q = "the shape has exactly four sides." Premise 1 gives us P → Q.
Identify the negation: Premise 2 tells us the shape does not have exactly four sides.
Apply modus tollens: Since P → Q is true and Q is false, P must be false.
Answer: The shape is not a square.
Why It Matters
Modus tollens appears throughout geometry proofs whenever you argue by contradiction or use the contrapositive. It is also a foundational rule in discrete mathematics and computer science, where automated theorem provers rely on it to eliminate possibilities.
Common Mistakes
Mistake: Confusing modus tollens with the fallacy of denying the antecedent. Students sometimes reason: P → Q, ¬P, therefore ¬Q.
Correction: Denying the antecedent is invalid. Modus tollens requires you to deny the consequent (¬Q), not the antecedent (¬P). Always check which part of the conditional you are negating.